Welcome to the Extended Sierpinski Problem
In 1962, John Selfridge discovered the Sierpinski number k = 78557, which is believed to be the smallest such number. The Sierpinski problem attempts to prove that it is, in fact, the smallest Sierpinski number. In 1976, Nathan Mendelsohn determined that the second provable Sierpinski number is the prime k = 271129. The prime Sierpinski problem attempts to prove that this is the smallest prime Sierpinski number.
Should both of these problems be solved, k = 78557 will be established as the smallest Sierpinski number, and k = 271129 will be established as the smallest prime Sierpinski number. However, this would not prove that k = 271129 is the second provable Sierpinski number. Since the prime Sierpinski problem is testing all prime k's for 78557 < k < 271129, all that's needed is to test the composite k's for 78557 < k < 271129. Thus, the extended Sierpinski problem is established.
As of November 25^{th}, 2021, there remain 8 composite k's for which no primes have been found. They are as follows:
91549, 131179, 163187, 200749, 209611, 227723, 229673, 238411
For a more detailed history and status of the extended Sierpinski problem, please visit Wilfrid Keller's The Sierpiński Problem: Definition and Status.
Primes found by PrimeGrid
202705*2^21320516+1 found by Pavel Atnashev on 25 November 2021  Official Announcement
99739*2^14019102+1 found by Brian D. Niegocki on 24 December 2019  Official Announcement
193997*2^11452891+1 found by Tom Greer on 3 April 2018  Official Announcement
161041*2^7107964+1 found by Martin Vanc on 6 January 2015  Official Announcement
211195*2^3224974+1 found by Ardo van Rangelrooij on 11 March 2013
94373*2^3206717+1 found by Jörg Meili on 10 March 2013
198677*2^2950515+1 found by Ardo van Rangelrooij on 23 October 2012
147559*2^2562218+1 found by Rodger Ewing on 27 March 2012
123287*2^2538167+1 found by Timothy D Winslow on 14 March 2012
250463*2^1316921+1 found by Rodger Ewing on 30 April 2010
219259*2^1300450+1 found by Lennart Vogel on 29 April 2010
154801*2^1305084+1 found by Rodger Ewing on 29 April 2010
98749*2^1045226+1 found by Rodger Ewing on 09 April 2010
107929*2^1007898+1 found by Brian Carpenter on 05 April 2010
85013*2^699333+1 found by Steve Martin on 25 March 2010
225679*2^620678+1 found by Lennart Vogel on 24 March 2010
187681*2^573816+1 found by Lennart Vogel on 23 March 2010
168587*2^545971+1 found by Steve Martin on 25 March 2010
208381*2^463068+1 found by Lennart Vogel on 22 March 2010
167957*2^417463+1 found by Brian Carpenter on 21 March 2010
185449*2^435402+1 found by Rodger Ewing on 21 March 2010
261203*2^354561+1 found by Lennart Vogel on 20 March 2010
227753*2^91397+1 found by Lennart Vogel 13 March 2010
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